It is easy to see that \(\widetilde{\Omega }_{\mathrm {c}}\)  is cyclic for \(\widetilde{\mathcal {M}}_{\mathrm {c},\kappa }^1\) . In order to see that \(\widetilde{\Omega }_{\mathrm {c}}\)  is separating
for \(\widetilde{\mathcal {M}}_{\mathrm {c},\kappa }\) , it is enough to prove that \(\widetilde{\mathcal {M}}_{\mathrm {c},\kappa }\)  and \(\widetilde{\mathcal {M}}_{\mathrm {c},\kappa }^1\)  commute.
This follows from Lemma REF , as in [1]}.
